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Gauss's Criterion

Let $p$ be an Odd Prime and $b$ a Positive Integer not divisible by $p$. Then for each Positive Odd Integer $2k-1< p$, let $r_k$ be

r_k\equiv (2k-1)b\ \left({{\rm mod\ } {p}}\right)

with $0<r_k<p$, and let $t$ be the number of Even $r_k$s. Then


where $(b/p)$ is the Legendre Symbol.


Shanks, D. ``Gauss's Criterion.'' §1.17 in Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 38-40, 1993.

© 1996-9 Eric W. Weisstein